Interferometric measuring systems can be used to monitor changes in the relative position of a measurement object based on an optical interference signal. For example, an interferometer generates the optical interference signal by overlapping and interfering a “measurement beam” that interacts with (e.g., reflects from) from the measurement object with a second beam, sometimes called a “reference beam” that does not interact with the measurement object. Changes in the relative position of the measurement object correspond to changes in the phase of the measured optical interference signal.
However, many interferometric measuring systems include nonlinearities such as what are known as “cyclic errors.” The cyclic errors can be expressed as contributions to the phase and/or the intensity of the measured interference signal and have a sinusoidal dependence on the change in an optical path difference between the measurement beam and the second beam. The cyclic errors can be produced as a result of “beam mixing” (where a portion of an input beam that nominally forms the reference beam propagates along a measurement path and/or vice versa) and imperfections in interferometer components (e.g., retro-reflectors and phase retardation plates). In some cases, non-harmonic cyclic errors (“Ce Nh”) are produced when a portion of the measurement beam and/or reference beam makes one or more passes through a partial measurement beam path. These cyclic errors have a frequency shift (i.e., the “Doppler” frequency) that is a non-integer multiple of the frequency difference between the components of the original input beam.
Another type of cyclic error includes the axis-dependent cyclic error (“CE Ad”). Axis-dependent cyclic errors may occur with changes in the relative position of the measurement object along multiple axes of motion. These cyclic errors have a cyclic error frequency ratio (i.e., ratio of cyclic error frequency shift to Doppler frequency) that is axis-dependent. For example, a different cyclic error frequency ratio may occur depending on whether the relative motion of the measurement object is along the X or Z direction. Both non-harmonic and axis-dependent cyclic errors can cause errors that are substantially larger in interferometric encoder systems compared to other types of interferometers. For example, a given cyclic error signal ratio can result in a larger position error (e.g., about three times as large) because the grating period of an encoder scale is generally coarser than the interference period of a comparable non-encoder based interferometer. Increased errors also can occur due to the reduced contrast (i.e., the ratio of time varying “AC” component of the interferometry signal relative to the background “DC” component of the interferometry signal) of interferometric encoder systems relative to other comparable interferometers.
In general, electronic compensation cannot correct for cyclic errors having non-harmonic or axis-dependent characteristics. Conventional methods of electronic compensation generate integer harmonics of the Doppler signal with simple trigonometric manipulations, and cannot compensate non-harmonic cyclic errors. Conventional methods of electronic compensation process signals from a single axis without consideration of other axes, and cannot compensate axis-dependent cyclic errors. Although filtering of CE Nh and CE Ad errors can be used in place of electronic compensation while the measurement object is in motion, filtering fails to correct for such errors during applications in which the speed of the relative motion is low, e.g., during alignment.